A rod of circular section is subjected to a shearing force on a plane perpendicular to its axis. What is the maximum shearing stress in terms of shearing force and rod diameter, if the rod is used as a beam with free ends and a central concentrated load, express the free length in terms of diameter for which the maximum shearing stress, due to shearing force is half the maximum direct stress ?
- AL = (3/2) D
- BL = (1/2) D
- CL = (5/2) D
- DL = (2/3) DCorrect
Solution
Concept
Max shear stress versus max bending stress in a circular beam
Formula
τmax = (4/3) V/A; σmax = M/Z
Step-by-step solution
1. Simply supported beam with central load W: V = W/2 and M = WL/4. 2. τmax = (4/3)(W/2)/(πD²/4) = 8W/(3πD²). 3. σmax = (WL/4)/(πD³/32) = 8WL/(πD³). 4. Set τmax = σmax/2: 8W/(3πD²) = 4WL/(πD³) ⇒ L = (2/3) D.
Explanation
For this relation the span is very short compared with the diameter (L = 0.67 D), i.e. a stocky beam in which shear is significant.
Why the other options are wrong
The other lengths would give a shear stress that is a different fraction of the bending stress.
Shortcut
τmax = 4V/3A; σmax = 8WL/πD³.
Common mistake
Using the average shear stress.
Correct answer: D) L = (2/3) D